Supervised classification of multidimensional and irregularly sampled signals

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Supervised classification of multidimensional and irregularly sampled signals

Alexandre Constantin, Mathieu Fauvel, Stéphane Girard, Serge Iovleff

To cite this version:

Alexandre Constantin, Mathieu Fauvel, Stéphane Girard, Serge Iovleff. Supervised classification of multidimensional and irregularly sampled signals. Statlearn 2019 - Workshop on Challenging problems in Statistical Learning, Apr 2019, Grenoble, France. pp.1. �hal-02092347�

Supervised classification of multidimensional and irregularly sampled signals.

Alexandre Constantin ¹ , Mathieu Fauvel ² , Stéphane Girard ¹ and Serge Iovleff ³

1

Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France

2

CESBIO, Université de Toulouse, CNES/CNRS/IRD/UPS/INRA, Toulouse, France

3

Laboratoire Paul Painlevé - Université Lille 1, CNRS, Inria, France

Introduction

Background:

Recent space missions, such as Copernicus Sentinel-2

^a

, provide high resolution Satellite Image Time Series (SITS) to study continental surfaces, with a very short revisit period (5 days for sentinel-2). In order to process such data, statistical models are regularly used [1, 2], which usually require a regular temporal sampling. However, for SITS, clouds and shadows (eg. figure from [3]), as well as the satellite orbite, an irregular temporal sampling is common.

Contribution:

A new statistical approach using Gaussian processes is proposed to classify irregularly sam- pled signals without temporal rescaling. Moreover, the model offers a theoretical framework to impute missing values such as cloudy pixels.

a

https://www.esa.int/Our_Activities/Observing_the_Earth/Copernicus/Sentinel-2

Model

Gaussian Processes (GP ) model:

Let S =

(y

_i

, z

_i

)

ⁿ

i=1

a set of multidimensional and irregularly sampled signals. A signal Y is modeled as a vector of p independent random processes T → R

^p

, with T = [0, T ]. The associated label is modeled by a discrete random variable Z taking its values in {1, . . . , C}. The model introduced here is based on two assumptions: 1) The coordinate processes Y

_b

, b ∈ {1, . . . , p} of Y are independent, 2) Each process Y

_b

is, conditionally to Z = c, a Gaussian process. Then

Y

_b

(t)|Z = c ∼ GP (m

_b,c

(t), K

_b,c

(t, s)),

where m

_b,c

: T → R

^p

is a mean function, and K

_b,c

a covariance kernel with hyperparameters θ

_b,c

. For example θ

_b,c

= {γ

_b,c²

, h

_b,c

, σ

_b,c²

} with

K

_b,c

(t, s) = γ

_b,c²

k(t, s|h

_b,c

) + σ

_b,c²

δ

_t,s

An irregularly sampled noisy signal y

_i

is observed on T

_i

time stamps {t

ⁱ₁

, . . . , t

ⁱ_T

i

} ∈ T and its bth coordinate is represented by a vector in R

^Ti

. We write y

_i,b

= [Y

_bⁱ

(t

ⁱ₁

), . . . , Y

_bⁱ

(t

ⁱ_T

i

)]

^T

, with

y

_i,b

|Z

_i

= c ∼ N

_Ti

µ

_i,b,c

, Σ

ⁱ_b,c

.

There µ

_i,b,c

= B

ⁱ_b

α

_b,c

is the sampled mean projected on a finite- dimensional space (B

ⁱ_b

is the fixed design matrix, α

_b,c

is the unknown vector of coordinates). Σ

ⁱ_b,c

is the matrix kernel K

_b,c

evaluations at {t

ⁱ₁

, . . . , t

ⁱ_T

i

}.

Estimation:

• α

_b,c

and θ

_b,c

are estimated by maximizing the log-likelihood,

− 1 2

X

i|Z_i=c

log

Σ

ⁱ

(θ

_b,c

)

+ (y

_i,b

− B

ⁱ_b

α

_b,c

)

^>

Σ

ⁱ

(θ

_b,c

)

⁻¹

(y

_i,b

− B

ⁱ_b

α

_b,c

).

• α

_b,c

is given by an explicit formula, while θ

_b,c

is computed thanks to a gradient technique.

Classification and Imputation of missing values

The assigned class is given by the MAP rule from the posterior probability

P (Z = c|y

_j

) = π ˆ

_c

Q

p

b=1

f

_Tj

y

_j

, B

^j_b

α ˆ

_b,c

, Σ

^j

(ˆ θ

_b,c

)

K

P

`=1

π ˆ

_`

Q

^p

b=1

f

_Tj

y

_j

, B

^j_b

α ˆ

_b,`

, Σ

^j

(ˆ θ

_b,`

) .

When the class is known to be c, the missing value at t

^∗

is estimated through the computation of conditional expectation.



 



 



Y ˆ

_b,cⁱ

(t

^∗

) =B

_bⁱ

(t

^∗

) ˆ α

_b,c

+ K

_b,c

(t

^∗

, t

_1:Ti

)

^>

Σ

ⁱ

(ˆ θ

_b,c

)

⁻¹

(y

_i,b

− B

_bⁱ

α ˆ

_b,c

) var( ˆ Y

_b,cⁱ

(t

^∗

)) =K

_b,c

(t

^∗

, t

^∗

)

− K

_b,c

(t

^∗

, t

_1:Ti

)Σ

ⁱ

(ˆ θ

_b,c

)

⁻¹

K

_b,c

(t

_1:Ti

, t

^∗

)

We also generalized this imputation when the class is unknown.

Validation (Synthetic data)

0 10 20 30 40 50

t: temporal instants 2

1 0 1 2 3 4 5 6

Y(t): Amplitude (1 band)

Example of two signals (dots) that belongs to two different classes Classification rate based on average time samples

n

_t

5 10 25 50 75

Acc

_exp

(%) 52.8 52.9 74.3 93.9 94.2 Acc

_sin

(%) 64.3 85.3 100 100 100

0 10 20 30 40 50

−2 0 2 4 6

Amplitude

Noisy observed signal Predicted signal

2 * Standard deviation Missing values

0 10 20 30 40 50

Time

−2 0 2 4 6

Amplitude

Imputation on two signals belonging to the same class.

Future work

We are now implementing the model for massive real data (Sentinel-2).

We are also working on a new model when the bands are correlated.

This work is supported by the French National Research Agency in the frame- work of the Investissements d’Avenir program (ANR-15-IDEX-02) and by the Centre National d’Etudes Spatiales (CNES).

[1] P. J. Brockwell and R. A. Davis. Time Series: Theory and Methods.

Springer-Verlag, Berlin, Heidelberg, 1986.

[2] C. K. Williams and C. E. Rasmussen. Gaussian processes for machine learning. the MIT Press, 2006.

[3] Sentinel hub blog. https://medium.com/sentinel-hub . Accessed:

2019-03-21.